L(s) = 1 | + (−1.41 + 1.41i)3-s + (1.41 + 1.41i)5-s − 4i·7-s − 1.00i·9-s + (1.41 + 1.41i)11-s + (1.41 − 1.41i)13-s − 4.00·15-s + 2·17-s + (1.41 − 1.41i)19-s + (5.65 + 5.65i)21-s − 4i·23-s − 0.999i·25-s + (−2.82 − 2.82i)27-s + (4.24 − 4.24i)29-s − 4.00·33-s + ⋯ |
L(s) = 1 | + (−0.816 + 0.816i)3-s + (0.632 + 0.632i)5-s − 1.51i·7-s − 0.333i·9-s + (0.426 + 0.426i)11-s + (0.392 − 0.392i)13-s − 1.03·15-s + 0.485·17-s + (0.324 − 0.324i)19-s + (1.23 + 1.23i)21-s − 0.834i·23-s − 0.199i·25-s + (−0.544 − 0.544i)27-s + (0.787 − 0.787i)29-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.427104703\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.427104703\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (1.41 - 1.41i)T - 3iT^{2} \) |
| 5 | \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + (-1.41 - 1.41i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-7.07 - 7.07i)T + 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (4.24 + 4.24i)T + 43iT^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9.89 - 9.89i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.41 - 1.41i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.07 - 7.07i)T - 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (4.24 - 4.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26596801300797928782725721098, −9.646216659172120929279571956175, −8.298887863837618844711182852651, −7.30701665261874746289978661546, −6.51572067217407206485432866094, −5.75580977862261704534476117709, −4.60051403689155093043264581036, −4.04885733417651694282974305745, −2.72603965828437921003271189657, −0.949978495414823838712843208652,
1.10078576728659743524075207070, 2.08258155784625936442467934634, 3.52195504667435645060392012954, 5.13313904663649993546507091065, 5.71825374667437010823562266658, 6.21706140665644193498539726895, 7.22701457878079858275664800472, 8.358257775398827819072167173389, 9.115936908234249311858173439330, 9.620850926648493260592906246203